Butterfly Effect

The butterfly effect refers to a concept that small causes can have large effects. Initially, it was used with weather prediction but later the term became a metaphor used in and out of science.

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Strange Attractors – The butterfly effect

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Edward Loren

In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. The name, coined by Edward Lorenz for the effect which had been known long before, is derived from the metaphorical example of the details of a hurricane (exact time of formation, exact path taken) being influenced by minor perturbations such as the flapping of the wings of a distant butterfly several weeks earlier. Lorenz discovered the effect when he observed that runs of his weather model with initial condition data that was rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.

The butterfly effect is exhibited by very simple systems. For example, the randomness of the outcomes of throwing dice depends on this characteristic to amplify small differences in initial conditions—the precise direction, thrust, and orientation of the throw—into significantly different dice paths and outcomes, which makes it virtually impossible to throw dice exactly the same way twice.

A plot of Lorenz’s strange attractor for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical systemthat, starting from any of various arbitrarily close alternative initial conditions on the attractor, theiterated points will become arbitrarily spread out from each other.

History

Chaos theory and the sensitive dependence on initial conditions were described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology.

In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.In 1963 Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow(the calculations were performed on a Royal McBeeLGP-30 computer).Elsewhere he stated:

One meteorologist remarked that if the theory were correct, one flap of a sea gull‘s wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.

Following suggestions from colleagues, in later speeches and papers Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.

The phrase refers to the idea that a butterfly’s wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly’s wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn’t. The flapping wing represents a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it’s also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.

The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.

Some scientists have since argued that the weather system is not as sensitive to initial condition as previously believed.David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations.

Illustration

These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of time in the Lorenz attractorstarting at two initial points that differ by only 10−5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the zcoordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t = 30.

An animation of the Lorenz attractor shows the continuous evolution.

Theory and mathematical definition

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical.

If M is the state space for the map , then displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d(. , .) such that and such that

for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent.

The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map:

where the initial condition parameter is given by . For rational , after a finite number of iterations maps into a periodic sequence. But almost all are irrational, and, for irrational , never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps folded within the range [0, 1].

Examples

The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example.

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem.Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al. presented a quantum algorithm to measure fidelity decay, which “measures the rate at which identical initial states diverge when subjected to slightly different dynamics”. They consider fidelity decay to be “the closest quantum analog to the (purely classical) butterfly effect”.Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.

The butterfly effect has also played a large role in many modern video games. There have been many instances of it being used, where a single/multiple choice(s) throughout gameplay may alter the entire ending of the game. A few examples are Heavy Rain, Beyond Two Souls, Until Dawn, and Life is Strange.

Two states differing by imperceptible amounts may eventually evolve into two considerably different states … If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible….In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.

His description of the butterfly effect followed in 1969. He was awarded the Kyoto Prize for basic sciences, in the field of earth and planetary sciences, in 1991, the Buys Ballot Award in 2004, and the Tomassoni Award in 2008. In his later years, he lived inCambridge, Massachusetts. He was an avid outdoorsman, who enjoyed hiking, climbing, and cross-country skiing. He kept up with these pursuits until very late in his life, and managed to continue most of his regular activities until only a few weeks before his death. According to his daughter, Cheryl Lorenz, Lorenz had “finished a paper a week ago with a colleague.” On April 16, 2008, Lorenz died at his home in Cambridge at the age of 90, having suffered from cancer.

Work

Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather patterns he began to realize that the weather patterns did not always behave as predicted. Minute variations in the initial values of variables in his twelve-variable computer weather model (c. 1960, running on an LGP-30 desk computer) would result in grossly divergent weather patterns.This sensitive dependence on initial conditions came to be known as the butterfly effect (it also meant that weather predictions from more than about a week out are generally fairly inaccurate).

Lorenz went on to explore the underlying mathematics and published his conclusions in a seminal work titled Deterministic Nonperiodic Flow, in which he described a relatively simple system of equations that resulted in a very complicated dynamical object now known as the Lorenz attractor.

Publications

Lorenz published several books and articles. A selection:

1955 Available potential energy and the maintenance of the general circulation. Tellus. Vol.7